Mitigating data gaps in the estimation of primaries by sparse inversion without data reconstruction

Transcription

1 Mitigating data gaps in the estimation of primaries by sparse inversion without data reconstruction Tim T.Y. Lin SLIM University of British Columbia

2 Talk outline Brief review of REPSI Data reconstruction in EPSI Eliminate explicit expression for missing data Field data example

3 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) true primary wavefield SRME-produced primary P o =P A(f)P o P P P o A(f) total up- going wavefield primary wavefield matching operator

4 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) true primary wavefield SRME-produced primary P o P A(f)PP SRMP P P o A(f) total up- going wavefield primary wavefield matching operator

5 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) X adaptive subtraction min A f kp A(f)PPk SRMP P P o A(f) total up- going wavefield primary wavefield matching operator

6 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) true primary wavefield SRME-produced primary P o =P A(f)P o P P P o A(f) total up- going wavefield primary wavefield matching operator

7 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME P=P o +A(f)P o P P P o A(f) total up- going wavefield primary wavefield matching operator

8 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME P=P o +A(f)P o P P o =QG A(f) = Q 1 P Q G total up- going wavefield down- going source signature primary impulse response

9 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME P=QG GP P Q G total up- going wavefield down- going source signature primary impulse response

10 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME P=QG GP Inversion objective: f(g, Q) = 1 2 kp (QG GP)k2 2

11 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) 0.4 recorded data predicted data from SRME P=QG GP 1 Inversion objective: Position (m) f(g, Q) = 1 2 kp (QG GP)k2 2

12 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME 0.4 P=QG GP Inversion objective: f(g, Q) = 1 2 kp (QG 1.4 GP)k Position (m)

13 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME 0.4 P=QG GP Inversion objective: f(g, Q) = 1 2 kp (QG 1.4 GP)k Position (m)

14 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME 0.4 P=QG GP Inversion objective: f(g, Q) = 1 2 kp (QG 1.4 GP)k Position (m)

15 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME 0.4 P=QG GP Inversion objective: f(g, Q) = 1 2 kp (QG 1.4 GP)k Position (m)

16 From SRME to Robust EPSI Based on Estimation of Primaries by Sparse Inversion (van Groenestijn and Verschuur, 2009) recorded data predicted data from SRME P=QG GP Inversion objective: f(g, Q) = 1 2 kp (QG GP)k2 2

17 From SRME to Robust EPSI Two ways to obtain the final primary wavefield Direct Primary Conservative Primary QG = P + GP Inversion objective: f(g, Q) = 1 2 kp (QG GP)k2 2

18 From SRME to Robust EPSI In time domain (lower- case: whole dataset in time domain) recorded data predicted data from SRME p=m(g, q) M(g, q) :=F t BlockDiag!1! nf [(q(!)i P) I]F t g Inversion objective: f(g, q) = 1 2 kp M(g, q)k2 2

19 Solving the EPSI problem Linearizations p=m(g, q) M q = M q g In fact it is bilinear: QG = P + GP M q g = M(g, q) M g q = M(q, g)

20 Solving the EPSI problem Linearizations p=m(g, q) M q = M q g Associated objectives: f q (g) = 1 2 kp M qgk 2 2 f g (q) = 1 2 kp M gqk 2 2

21 Solving the EPSI problem Do: g k+1 = g k + S(rf qk (g k )) q k+1 = q k + rf gk+1 (q k ) Gradient sparsity S : pick largest elements per trace

22 Robust EPSI L1-minimization approach to the EPSI problem [Lin and Herrmann, 2013 Geophysics] While kp M(g k, q k )k 2 > determine new k from the Pareto curve g k+1 = arg min g kp M qk gk 2 s.t. kgk 1 apple k q k+1 = arg min q kp M gk+1 qk 2

23 Choosing Tau from the Pareto curve Look at the solu:on space and the line of op:mal solu:ons (Pareto curve) minimize kxk 1 subject to kax bk 2 apple ka H rk 1 krk 2 g k k solving LASSO via projected gradient g k+1 minimize kax bk 2 subject to kxk 1 apple k k+1 feasible solu:on with smallest x 1

24 Inverting for unknown data

25 Robust EPSI Inverting for unknown data P(G k+1, R k+1 ):= (I + G k+1 ) H (R k+1 ) While kp k M(g k, q k )k 2 > determine new k from the Pareto curve g k+1 = arg min g kp k M qk gk 2 s.t. kgk 1 apple k q k+1 = arg min q kp k M gk+1 qk 2 p k+1 = p k + p(g k+1, q k+1, p k )

26 Robust EPSI Inverting for unknown data Data changes every iteration! P(G k+1, R k+1 ):= (I + G k+1 ) H (R k+1 ) While kp k M(g k, q k )k 2 > determine new k from the Pareto curve g k+1 = arg min g kp k M qk gk 2 s.t. kgk 1 apple k q k+1 = arg min q kp k M gk+1 qk 2 p k+1 = p k + p(g k+1, q k+1, p k )

27 Trace Mask Masking operator K

28 Trace Mask Bisects wavefield data to unknown/ uncertain traces (i.e., near offset)

29 Trace Mask Masking operator K Time domain: Frequency slices: Kp K P

30 Trace Mask Complement of Masking operator K c Time domain: Frequency slices: K c p K c P

31 Bisected data variables P 0 + P 00 = P Known data traces: P 0 := K P Unknown data traces: P 00 := K c P

32 Bisected data variables P 0 + P 00 = P Known data traces: P 0 := K P Unknown data traces: P 00 := K c P = K c GQ + RGP 0 + RGP 00

33 Bisected data variables P 0 + P 00 = P Known data traces: P 0 := K P Unknown data traces: P 00 := K c P = K c GQ + RGP 0 + RGP 00 Highly dependent on G

34 Bisected data variables P 0 + P 00 = P Known data traces: P 0 := K P Unknown data traces: P 00 := K c P = K c GQ + RGP 0 + RGP 00 Recursively defined

35 Modify the modeling operator M(G, Q; P 0 )=GQ + RGP 0 + RGP 00

36 Modify the modeling operator fm(g, Q; P 0 )=GQ + RGP 0 + K RGK c (GQ + RGP 0 ) + O(G 3 )

37 Modify the modeling operator fm(g, Q; P 0 )=GQ + RGP 0 2nd Order autoconvolution term + K RGK c (GQ + RGP 0 ) + O(G 3 )

38 Modify the modeling operator fm(g, Q; P 0 )=K GQ + RGP 0 + RGK c (GQ + RGP 0 ) + O(G 3 )

39 Modify the modeling operator fm(g, Q; P 0 )=K GQ + RGP 0 2nd Order autoconvolution term 3rd Order autoconvolution term + RGK c (GQ + RGP 0 ) + RGK c (RGK c (GQ + RGP 0 )) + O(G 4 ) 1X :=K (RGK c ) n (GQ + RGP 0 ). n=0

40 Convergent sum 1X K P =K (RGK c ) n (GQ + RGP 0 ) n=0 = K (I RGK c ) 1 (GQ + RGP 0 )

41 Convergent sum 1X K P =K (RGK c ) n (GQ + RGP 0 ) n=0 = K (I RGK c ) 1 (GQ + RGP 0 ) Verifies the validity of the expression

42 What these terms look like

43 Position (km) Position (km) Position (km) Data with missing traces P Exact multiples - GP

44 Position (km) Position (km) Position (km) Missing contributions Exact multiples - GP

45 Position (km) Position (km) Position (km) Missing contributions 2nd order term 3rd order term

46 Modify the modeling operator fm(g, Q; P 0 )=K GQ + RGP 0 2nd Order autoconvolution term 3rd Order autoconvolution term + RGK c (GQ + RGP 0 ) + RGK c (RGK c (GQ + RGP 0 )) + O(G 4 ) 1X :=K (RGK c ) n (GQ + RGP 0 ). n=0

47 Position (km) Position (km) Position (km) Missing contributions 2nd order term 3rd order term

48 Position (km) Position (km) Missing contributions 2nd order term + 3rd order

49 Total missing contrib Position (km)

50 Modeled with two terms Position (km)

51 Solution strategy

52 Robust EPSI Inverting for unknown data P(G k+1, R k+1 ):= (I + G k+1 ) H (R k+1 ) While kp k M(g k, q k )k 2 > determine new k from the Pareto curve g k+1 = arg min g kp k M qk gk 2 s.t. kgk 1 apple k q k+1 = arg min q kp k M gk+1 qk 2 p k+1 = p k + p(g k+1, q k+1, p k )

53 Robust EPSI Accounting for unknown data with G While kp f M(gk, q k )k 2 > determine new k from the Pareto curve g k+1 = arg min g kp f M qk gk 2 s.t. kgk 1 apple k q k+1 = arg min q kp f M gk+1 qk 2

54 Strategy 1: Re-linearization Use G from previous iter in higher-order terms While kp f M(gk, q k )k 2 > determine new k from the Pareto curve g k+1 = arg min g kp f M qk gk 2 s.t. kgk 1 apple k q k+1 = arg min q kp f M gk+1 qk 2 use g k in these operators

55 Strategy 2: Modified Gauss-Newton Obtain Jacobian using G from previous iter While kp f M(gk, q k )k 2 > determine new k from the Pareto curve g k+1 = g k + arg min g kr (gk,q k ) f M gk 2 s.t. k gk 1 apple k q k+1 = arg min q kp f M gk+1 qk 2

56 Field data example North Sea dataset

57 0 CMP position (km) Offset (km) North Sea dataset 100m near-offset regularized to 12.5m dx and 4km fixed-spread from streamer 4ms sampling 2.0 Data Shot

58 0 CMP position (km) CMP position (km) NMO stack Parabolic Radon Interp Conservative primary Multiple

59 0 CMP position (km) CMP position (km) NMO stack Re-linearization Using 3rd Order terms Conservative primary Multiple

60 0 Offset (km) Offset (km) Direct primary Original shots

61 0 Offset (km) Offset (km) Direct primary shot Interpolated shot

62 0 CMP position (km) CMP position (km) NMO stack Re-linearization Using 2nd Order terms Conservative primary Multiple

63 0 CMP position (km) CMP position (km) NMO stack Difference plots Radon interp - Re- linearization 2nd order Re- linearization 3rd - 2nd order

64 0 CMP position (km) CMP position (km) NMO stack Modified Gauss-Newton Using 3rd Order terms Conservative primary Multiple

65 0 CMP position (km) CMP position (km) NMO stack Re-linearization Using 3rd Order terms Conservative primary Multiple

66 0 CMP position (km) Offset (km) North Sea dataset 100m near-offset regularized to 12.5m dx and 4km fixed-spread from streamer 4ms sampling 2.0 Data Shot

67 0 CMP position (km) CMP position (km) CMP position (km) Difference GN 3rd Order Multiple Re- lin. 3rd Order Multiple

68 Discussion and summary Able to obtain an approximate forward operator with incomplete data without dependence on the missing traces Inversion should be more stable by not changing the data at each iteration Further work: More careful study needed in comparison to explicit data inversion More detailed comparison of algorithms What does the infilled G have to look like? Is it always the physical Green s function?

69 Acknowledgements PGS for field dataset This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant (22R81254) and the Collaborative Research and Development Grant DNOISE II ( ). This research was carried out as part of the SINBAD II project with support from the following organizations: BG Group, BGP, BP, CGG, Chevron, ConocoPhillips, ION, Petrobras, PGS, Statoil, Total SA, Sub Salt Solutions, WesternGeco, and Woodside.